Even among scientists, it is assumed that mathematics plays a secondary role: It is thought of as a toolkit, not the research itself. A biologist, say, would collect data and then try to build a mathematical model fitting these data, perhaps with some help from a mathematician. While this is an important mode of operation, mathematics actually plays a much bigger role in science: It enables us to make groundbreaking leaps that we couldn't make otherwise.

For example, Albert Einstein wasn’t trying to fit any data into a mathematical model when he realized that gravity causes our space-time to curve. In fact, there was no such data. No one could even imagine that our space is curved; everyone “knew” that our world was flat! (Note that I am not talking here about the Earth being curved which of course had been known for centuries; I am talking about the four-dimensional space-time we inhabit being curved.) How did Einstein come up with this far-out idea? He tried to generalize his special relativity theory to allow acceleration, using his insight that gravity and acceleration have the same effect. And Einstein followed in the footsteps of a mathematician, Bernhard Riemann, who laid the foundations of the theory of curved spaces 50 years earlier. It was math that gave the answer.

The human brain is wired in such a way that we simply cannot imagine curved shapes of dimension greater than two. Like all of us, Einstein could not possibly visualize a curved universe. He could describe it only using the language of mathematics. A subsequent experiment proved Einstein right. It turned out that our universe was indeed curved: A ray of light does not travel along a straight line, but bends passing near a star, as if pulled by an invisible force—a startling revelation.

The prediction of the Higgs boson is another beautiful example of mathematics driving progress in natural science. In the 1960s, physicists struggled with the fact that an attractive mathematical theory governing the behavior of elementary particles gave a nonsensical answer: It predicted massless particles that no one had seen. What we now know as the Higgs boson solved this problem. Inserted into the equations in just the right way, it gives particles their masses. The rest is history.

The Higgs boson was the last missing piece of the Standard Model, and its experimental discovery was the end of an era. Most physicists agree that the Standard Model is not the ultimate theory of the quantum world: For one thing, it gives us no clue about the mysterious dark matter that takes up over 80 percent of the total matter in the universe. We need new ideas to go beyond the Standard Model, but it’s quite possible that no new particles or phenomena will be discovered at the LHC within its energy range. Should we build another accelerator to reach energies even higher? A bigger accelerator would cost much more than the 10 billion dollars the LHC had cost and take a long time to build, with no guarantee that any new physics will be found from it. It may well be that the next breakthrough in quantum physics will again come from mathematics, just as it did through the work of Englert, Higgs, and others that we celebrate this week.

Upon hearing that a telescope at the Mount Wilson Observatory was needed to study the cosmos, Albert Einstein's wife Elsa remarked: “Well, my husband does that on the back of his envelope.” Experiment is the ultimate judge of a theory, and that’s why we do need expensive and sophisticated machines. But the amazing fact is that scientists like Einstein and Higgs have used the most abstract mathematical knowledge to unlock the deepest secrets of the universe.

Charles Darwin wrote in his autobiography: “I have deeply regretted that I did not proceed far enough at least to understand something of the great leading principles of mathematics, for men thus endowed seem to have an extra sense.” Mathematics is not about studying boring and useless equations: It is about accessing a new way of thinking and understanding reality at a deeper level. It endows us with an extra sense and enables humanity to keep pushing the boundaries of the unknown.

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